| L(s) = 1 | − 2-s − 4-s − 5-s + 3·8-s − 2·9-s + 10-s − 4·11-s − 16-s + 4·17-s + 2·18-s − 4·19-s + 20-s + 4·22-s − 2·23-s − 2·25-s − 6·29-s − 8·31-s − 5·32-s − 4·34-s + 2·36-s + 4·37-s + 4·38-s − 3·40-s − 4·41-s − 4·43-s + 4·44-s + 2·45-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.06·8-s − 2/3·9-s + 0.316·10-s − 1.20·11-s − 1/4·16-s + 0.970·17-s + 0.471·18-s − 0.917·19-s + 0.223·20-s + 0.852·22-s − 0.417·23-s − 2/5·25-s − 1.11·29-s − 1.43·31-s − 0.883·32-s − 0.685·34-s + 1/3·36-s + 0.657·37-s + 0.648·38-s − 0.474·40-s − 0.624·41-s − 0.609·43-s + 0.603·44-s + 0.298·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18080 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18080 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 113 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T - 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 76 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 88 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 22 T + 246 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.2487816915, −15.7608732300, −15.0968647829, −14.6320907105, −14.3952422624, −13.5852960433, −13.2193926937, −12.8258304104, −12.2659148028, −11.5775217170, −11.1180744664, −10.6182264748, −10.0736371150, −9.69041950471, −8.95846457415, −8.45995724701, −8.05965899946, −7.50398637908, −7.06687532480, −5.80906664538, −5.58538939642, −4.71555215050, −3.95930694063, −3.15960680451, −1.93890918093, 0,
1.93890918093, 3.15960680451, 3.95930694063, 4.71555215050, 5.58538939642, 5.80906664538, 7.06687532480, 7.50398637908, 8.05965899946, 8.45995724701, 8.95846457415, 9.69041950471, 10.0736371150, 10.6182264748, 11.1180744664, 11.5775217170, 12.2659148028, 12.8258304104, 13.2193926937, 13.5852960433, 14.3952422624, 14.6320907105, 15.0968647829, 15.7608732300, 16.2487816915
